The mass of part of a wire is [tex]$x(1+\sqrt{x})$[/tex] kilograms, where [tex]$x$[/tex] is measured in meters from one end of the wire. Find the linear density of the wire when [tex]$x=36 m$[/tex].
Answer (1)
The mass of the wire is given by m ( x ) = x ( 1 + x ) .
Find the linear density by taking the derivative of the mass function with respect to x : ρ ( x ) = d x d m = 1 + 2 3 x .
Evaluate the linear density at x = 36 : ρ ( 36 ) = 1 + 2 3 36 = 10 .
The linear density of the wire at x = 36 meters is 10 kg/m .
Explanation
Problem Setup We are given the mass of a part of a wire as a function of the distance x from one end of the wire: m ( x ) = x ( 1 + x ) kilograms. We need to find the linear density of the wire at x = 36 meters.
Linear Density as Derivative The linear density ρ ( x ) is the rate of change of mass with respect to length, which is the derivative of the mass function m ( x ) with respect to x . So, we need to find d x d m .
Finding the Derivative First, let's rewrite the mass function as m ( x ) = x + x 3/2 . Now, we can find the derivative with respect to x :
d x d m = d x d ( x + x 3/2 ) = 1 + 2 3 x 1/2 So, the linear density function is ρ ( x ) = 1 + 2 3 x .
Evaluating at x=36 Now, we need to evaluate the linear density function at x = 36 meters: ρ ( 36 ) = 1 + 2 3 36 = 1 + 2 3 ( 6 ) = 1 + 9 = 10 Therefore, the linear density of the wire at x = 36 meters is 10 kg/m.
Final Answer The linear density of the wire at x = 36 m is 10 kg/m.
Examples
Imagine you're designing a suspension bridge, and you need to determine the density of the cable at a specific point to ensure it can withstand the tension. By knowing the mass distribution function along the cable, you can find the linear density at any point, which is crucial for structural integrity. This problem demonstrates how calculus helps engineers make informed decisions about material properties in real-world applications.
